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A Slope Problem Involving a Billiards Table

Date: 10/19/2004 at 21:09:29
From: Christie 
Subject: The Billiard Problem

Suppose that a square billard table has corners at (0,0),(1,0),(1,1),
(0,1). A ball leaves the origin along a line with slope s. If the ball
reaches a corner, it stops or falls in.  Whenever it hits the side of
the table not at a corner, it continues to travel on the table, but
the slope on the line is multiplied by -1. Does the ball reach a corner?

The solution to the problem is when s = 3/5.  I do not understand why.



Date: 10/20/2004 at 11:41:00
From: Doctor Douglas
Subject: Re: The Billiard Problem

Hi, Christie.

Thanks for writing to the Math Forum.

I think what is meant is that if the trajectory passes through one of 
the four corners, then the ball stops.

You can analyze this problem by considering the grid of all points
(x,y) where x and y are nonnegative integers.  If the initial
trajectory from the origin hits one of the points in the entire 
grid, then the ball will stop there (if it hasn't already stopped).

The case of initial slope s = 3/5 will certainly hit the grid point
at (x,y) = (5,3), which is point A in the figure below.  You can
verify that it doesn't hit any earlier.
  
  y
  |   |   |   |   |   |   |
  3 - . - . - . - . - A - . -
  |   |   |   |   |   |   |
  2 - . - . - . - . - . - . -
  |   |   | K |   |   |   |
  1 - . - . - . - . - . - . -
  |   |   |   |   |   |   |
  O - 1 - 2 - 3 - 4 - 5 - 6 - x

The line from O to A passes through a number of lines (I calculate
six).  If it passes through a vertical line, the ball bounces on one 
of the left/right walls.  If it passes through a horizontal line, it 
bounces on either the top or bottom walls.  By drawing the line from O 
to A, you can track the entire history of how many times the ball 
bounces and exactly where it goes.  For example, you can convince 
yourself that the ball passes through the exact middle of the table at 
point K, halfway through its trajectory. 

Many other trajectories lead to the ball hitting one of the corners.
In fact, any slope s = p/q that is rational will cause it to hit a
corner [at (q,p), if not earlier].  Slopes s that are irrational (such 
as pi, or sqrt(2) will allow the ball to bounce around the table 
forever.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Coordinate Plane Geometry

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